Packing spanning arborescences with extra large one

Abstract

The celebrated Nash-Williams and Tutte's theorem states that a graph $G=(V, E)$ contains $k$ edge disjoint spanning trees if and only if $\nu_{f}(G) \geq k$, where $$\nu_{f}(G):=\min_{|\mathcal{\mathcal{P}}|>1, \text{$\mathcal{P}$ is a partition of $V(G)$}}\frac{|E( \mathcal{P})|}{|\mathcal{P}|-1}.$$ Inspired by the NDT theorem as structural explanations for the fractional part of Nash-Williams' forest decomposition theorem, Fang and Yang extended Nash-Williams and Tutte's theorem and proved that if $\nu_{f}(G) > k+ \frac{d-1}{d}$, then $G$ contains $k$ edge disjoint spanning trees and another forest $F$ with $ |E(F)|> \frac{d-1}{d} (|V(G)|-1)|$, and if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. In this paper, we give a digraphic version of their result; however, the mixed graphic version remains open.